Universal credit migration letter issue

They do not ask you to provide that when filling out the UC claim form.

There should be a section for that yes.

It allegedly speeds up confirmation of ESA status, but from a few I've advised since then I don't think the system is working well. I still advise people to allow several weeks.

Transitional Protection is automatically calculated, sometimes it isn't done in time for the first statement.

If that happens and you feel you are entitled to TP (not everyone qualifies), you can add a Journal note chasing it.